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A note on some discrete valuation rings of arithmetical functions

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*Emil D. Schwab and Gheorghe Silberberg*

**Address.** E.D.Schwab, Department of Mathematics, University of Oradea,
str. Armatei Romane nr. 5, 3700 Oradea, ROMANIA
and

Department of Mathematical Sciences, University of Texas at El Paso,
El Paso, Texas, 79968-0514, USA

Gheorghe Silberberg, Department of Mathematics, University of Timisoara,
bd. V. Parvan nr. 4, 1900 Timisoara, ROMANIA

**E-mail:** c98sig01@student.ceu.hu

**Abstract.** The paper studies the structure of the ring A of arithmetical
functions, where the multiplication is defined as the Dirichlet convolution.
It is proven that A itself is not a discrete valuation ring, but a certain
extension of it is constructed,this extension being a discrete valuation
ring. Finally, the metric structure of the ring A is examined.

**AMSclassification.** 13F30

**Keywords.** Discrete valuation ring, arithmetical function, Dirichlet
convolution